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Full text of "Treatise On Analysis Vol-Ii"

18    CANONICAL DECOMPOSITIONS OF A MEASURE        211

to obtain an increasing sequence of sets Am <= X such that ^(X  Am) -> 0 as m -> oo

n

and in which the partial sums sn(x) =]T \fk(x)\2 tend uniformly to +00. Then remark

fc = i

that the series with general term \fn(x)\2lsn(x) is divergent almost everywhere (Section
5.3, Problem 6).

(c) Under the same hypotheses show that for almost all XQ e X there exists a function
g e ^c(X, p,) (depending on x0) such that the series with general term (g \fn)fn(xQ) has
sum equal to 4- co (use Problem 23 of Section 12.16).

6.    (a)    Let 6 : R->R be increasing and continuous on the right. Show that there exists
a unique positive measure v on R such that v(\a, b])=6(b)~ 6(a) for every half-open
interval ]a,b]. (This measure is called the "Stieltjes measure defined by 6", and we

write   fdB in place of   fdv.) Conversely, every positive measure on R may be obtained

in this way, and two functions 0Z, 02 on R (both of them increasing and continuous
on the right) define the same measure if and only if 92 ~~ &i is constant. Under what
conditions is the measure v diffuse? What is then the image of v under 0?

(b)    Let K be the Cantor set (Section 4.2, Problem 2). Show that there exists a diffuse
positive measure v on R with support K and total mass equal to 1 (Section 4.2, Problem
2(d)). Deduce that there exists on R a diffuse positive measure, disjoint from Lebesgue
measure, with support equal to I = [0, 1 ]. (In each component interval J of I  K
choose a measure proportional to the image of v under an affine linear mapping of
I onto J; then proceed by induction.)

(c)    Deduce from (a) that if ^ is a positive measure on R+ and/a function belonging

to &ioc, c(R +  /*) anc* (O a dense sequence in R such that           fdu = 0 for all n,

J[0, tn]

then/is /z-negligible.

7.    Let X be a compact space and /x a diffuse positive measure on X with total mass equal
to 1.

(a)    Construct a family (U(O)o^r^i of /^-quadrable open sets, such that U(0) ~ 0,
U(l) = X, U(0 c U(O whenever / < /', and j*(U(0) * for all t. (First define the
U(0 for t of the form /c/2", by induction on n. Use Problem 7(d) of Section 13.9 to
prove that if V, W are two quadrable open sets in X such that V c: w and /x( V) < jn(VV),
then there exists a quadrable open set U such that V c u <= 0 c W and

i/i(W - V)< JK(U - V) < J/x(W - V).

Also use the result of Problem 3(b).)

(b)    Deduce from (a) that there exists a continuous mapping TT of X onto I = [0,1]
such that the image of /x under TT is Lebesgue measure A on I.

(c)    Show that there exists a /^-negligible subset N of X, a A-negligible subset M of I
and a homeomorphism TTO of I  M onto X  N such that, for every A-measurable
subset A of I  M, the set TTO(A) is /x-measurable and of measure JJL(TTQ(A))  A(A).
(Use Problem 7(d) of Section 13.9 to show that for each integer n> 0 there exists a
finite partition of X consisting of a /^-negligible set and quadrable open sets of diameter
<l/n (with respect to a distance defining the topology of X) and of measure gl/w.
Proceeding by induction on n and passing to the limit, obtain a homeomorphism/
of I  D onto a subset of X of measure 1, where D is a denumerable subset of I.)

8.    Let X, Y be two compact spaces, U a continuous linear mapping of the Banach
space tfR(X) into the Banach space #n(Y), and 7r:X->Y a continuous  mapping.                                              fc=l